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On Convergence of the Immersed Boundary Method for Elliptic Interface Problems

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On Convergence of the Immersed Boundary Method for Elliptic Interface Problems On convergence of the immersed boundary method for elliptic interface problems Zhilin Li∗ January 26, 2012 Abstract Peskin's Immersed Boundary (IB) method is one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskin's IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the L1 norm. But almost no rigorous proof could be found in the literature until recently [14] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a log h factor of the IB method for elliptic interface problems essentially without the boundary condition restrictions. The results should be applicable to the IB method for many different situations involving elliptic solvers for Stokes and Navier-Stokes equations. keywords: Immersed Boundary (IB) method, Dirac delta function, convergence of IB method, discrete Green function, discrete Green's formula. AMS Subject Classification 2000 65M12, 65M20. 1 Introduction Since its invention in 1970's, the Immersed Boundary (IB) method [15] has been applied almost everywhere in mathematics, engineering, biology, fluid mechanics, and many many more areas, see for example, [16] for a review and references therein. The IB method is not only a mathematical modeling tool in which a complicated boundary condition can be treated as a source distribution but also a numerical method in which a discrete delta function is used. The IB method is robust, simple, and has been applied to many problems. ∗Center for Research in Scientific Computation (CRSC) and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA 1 It is widely believed that Peskin's IB method is only first order accurate in the L1 norm. How- ever, there was almost no rigorous proof in the literature until recently [14], in which the author has proved the first order accuracy of the IB method for the Stokes equations with a periodic boundary condition. The proof is based on some known inequalities between the fundamental solution and the discrete Green function with a periodic boundary condition for Stokes equations. In [4], the author showed that the pressure obtained from IB method has O(h1=2) order of convergence in the L2 norm for a 1D model. In [19, 20], the authors designed some level set methods based on discrete delta functions. With suitable quadrature formulas in the integral form using the Green functions, the authors show that their approach can get expected accuracy. However, there are few theoretical proofs on the IB method for elliptic interface problems with general boundary conditions. This is the main motivation of this paper. One difficult is that there is little known estimates between the fundamental solution and the discrete Green function with Dirichlet or other boundary conditions on rectangular domains. Compared with the case of periodic boundary conditions where there are existing estimates between the discrete Green function and the continuous one [8], there is almost none for Dirichlet and other boundary conditions. The main goal of this paper is to provide a con- vergence proof for the IB method for elliptic interface problems with Dirichlet boundary conditions. We will show that with commonly used discrete delta functions that satisfy the zeroth moment condition and first order interpolation property, the IB method is indeed first order convergent in the L1 norm with a log h factor. The key in our proof is to establish a connection between the dis- crete Green function and the continuous one. Our proof is essentially independent of the boundary conditions and it is valid in 1D, 2D, and 3D cases. The result should be applicable for many IB methods involving Stokes and Navier-Stokes solvers. 2 Proof of the convergence of the IB method in 1D We will give a proof for the 1D model, u00 = c δ(x α) 0 < x < 1; 0 < α < 1; u(0) = u(1) = 0; (1) − first in this section. Note that the analytic solution to the equation above is cx (1 α) if x α; u(x) = − − ≤ (2) ( cα (1 x) otherwise: − − Given a uniform Cartesian grid x = ih, i = 0; 1; ; n, h = 1=n, the IB method leads to the i · · · following system of linear equations, Ui 1 2Ui + Ui+1 − − = cδ (x α) ; i = 1; 2; ; n 1; (3) h2 h i − · · · − where U is the finite difference approximation of the solution u(x ), and δ (x α) is a discrete delta i i h i − function applied to the grid point xi. In the matrix-vector form, the above finite difference equations can be written as AhU = F, where Ah is the tri-diagonal matrix formed by the discrete Laplacian. 2 It is well known that A is a symmetric positive definite matrix (SPD) and diagonally dominant. − h Note that, a discrete delta function has a compact support in the neighborhood of interface, that is, δ (x) = 0 only if x W h, where W is a constant. Commonly used discrete delta functions include h 6 j j ≤ the hat discrete delta function (δhat(x) with W = 1): (h x )=h2; if x < h, hat − j j j j δh (x) = (4) ( 0; if x h, j j ≥ and Peskin's original discrete cosine delta function (δcosine(x) with W = 2) 1 (1 + cos (πx=2h)) ; if x < 2h; δ (x)cosine = 4h j j (5) h 8 0; if x 2h: < j j ≥ see for example, [13]. Note that, when: we use the hat delta function, the result is the same as that of the IIM for the simple model. The solution to the finite difference equations is the same as true solution if there are no round-off errors, that is u(xi 1) 2u(xi) + u(xi+1) hat − − = cδ (x α) ; i = 1; 2; ; n 1; (6) h2 h i − · · · − see for example, [3, 13]. But this is not the case for other discrete delta functions. We define the error vector as E = E , where E = u(x ) U . The local truncation error is f ig i i − i defined as T = T , f ig u(xi 1) 2u(xi) + u(xi+1) T = − − cδ (x α) : (7) i h2 − h i − With the definition, we have AhU = F, Ahu = F + T, and therefore AhE = T. For the hat discrete delta function, we have T = 0 for all i's for the simple model. For the cosine discrete delta function j ij or other discrete delta functions, generally we have T = O(1=h) for a few grid points neighboring j j j the interface α, see Table 1 on page 19. So the interesting question is: Why is the IB method still first order accurate, that is, E = O(h)? To answer this question, we first introduce the following k k1 lemma. Lemma 2.1. Let Ahy = ek and y0 = yn = 0, where ek is the k-th unit base vector, then hxi (1 xk) if i k; yi = − − ≤ (8) ( hxk (1 xi) otherwise: − − The significance of this lemma is that the solution is order h smaller than the concentrated source. Proof: We note the following identity e A y = h k = h δhat e : (9) h h h k 3 For this simple case, the IB method using the hat discrete delta function is identical to the IIM, see [13]. Thus from the Immersed Interface Method, see [3, 13], we know that y is the exact discrete solution at the grid points of the following boundary value problem u00 = h δ(x x ); 0 < x < 1; u(0) = u(1) = 0; (10) − k whose solution is hx (1 xk) if x xk; y(x) = − − ≤ (11) ( hxk (1 x) otherwise: − − This completes the proof. Note that y(x) h. From this lemma, we have the following corollary. j j ≤ Corollary 2.2. Let Ahy = r, then y(x) h W r , where W is the number of non-zero compo- j j ≤ k k1 nents of r. The proof is straightforward from (11) and the fact that 0 x 1 and 0 1 x 1. Notice ≤ ≤ ≤ − ≤ that for a discrete delta function, it should satisfy at least the zeroth moment equation, see [3], that is δ (x α) = 1; (12) h i − i X corresponding to the continuous case δ (x α)dx = 1. Now we are ready to prove the main h − theorem. R Theorem 2.3. Let u(x) be the solution to (1) and U is the solution obtained from the immersed boundary method (3) using a discrete delta function δh(x) for (1). Then U is first order accurate, that is E C¯h; (13) k k1 ≤ where C¯ is a constant. Proof: We can decompose the local truncation error into two groups T = Treg + Tirreg; (14) where Treg = 0 corresponds to the local truncation errors at regular grid points where δh(xi k k1 − α) = 0 and the true solution is piecewise linear. Note that, we have n 1 − reg irreg irreg Ti = T + T = O + T : (15) i=1 X X X X On the other hand, we also have u(xi 1) 2u(xi) + u(xi+1) hat − − = cδ (x α) (16) h2 h i − 4 since the finite difference method using the discrete delta function gives the exact solution at all the grid points. Thus we have n 1 − u(xi 1) 2u(xi) + u(xi+1) T = − − cδ (x α) i h2 − h i − i=1 X (17) n 1 n 1 − − = cδhat(x α) cδ (x α) = 0; h i − − h i − i=1 i=1 X X irreg irreg from the zeroth moment equation (12).
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